# Function properties

* A set is a collection of objects, called its elements.
* A class is a membership-restricted special set.
* A type is a class of objects, called values.
* A relation is a special set (class).

A function is a

* set
* set of ordered pairs
* relation

relation is a set type is a set relation is a set of ordered pairs

function is a set function is a relation function is a set of ordered pairs function is a type (function type)

## Properties of functions

A function is a special set.\
A function is a special type of relation.\
A type is a set.\
A function is a mapping between two sets (types).\
A relation is a set of ordered pairs (a,b).

A function is defined in terms of sets:\
a function between two sets A and B\
(or a function on a set A, in case A = B),\
is a special type of relation between A and B;

Any binary relation between two sets `A` and `B` is a set, `𝓡 = { (a,b) | a ∈ A ∧ b ∈ B }`, of ordered pairs, `(a,b)` such that the first component, `a`, is from the domain and the second, `b`, is from the codomain.

`a𝓡b` is the notation meaning that `a` is 𝓡-related to `b`; another notation, `(a,b) ∈ 𝓡`, means the same.

A function `f : ℝ → ℝ` with `∀x,y ∈ ℝ` is

* *order-preserving*   :  (x <= y) -> (f x <= f y)
* *metric-preserving*  :   |x − y| -> |f x − f y|
* *addition-preserving*: f (x + y) -> f x + f y

A function `f : ℝ → ℝ` with `∀x,y ∈ ℝ` (is it ℝ or any set?) is *monotonic* if the elements `x` and `y` are covariant and the results `f x` and `f y` stay covariant. For example, `(x <= y) -> (f x <= f y)`. A function is *non-monotonic* is the relation is reversed or contravariant, e.g. `(x <= y) -> (f x > f y)`. Some functions are neither or both; e.g. `f : ℝ → ℝ, f x = x²` is neither; it is non-monotonic for `x ∈ (-1,0) ∨ x ∈ (0,1)` and monotonic for other values.

| x     | x²     |
| ----- | ------ |
| -2.0  | 4      |
| -1.5  | 2.25   |
| -0.1  | 0.01   |
| -0.01 | 0.0001 |
| 0     | 0      |
| 0.01  | 0.0001 |
| 0.1   | 0.01   |
| 0.4   | 0.16   |
| 0.5   | 0.25   |
| 2     | 4      |